Fundamentals of Statistical Testing
Lecture 02
PSAs
Make sure you can see all the channels on Discord:
Analysing Data Roadmap
Speaking Stats
Learning to think in statistical terms is a skill just like:
Drawing/art/music
Weightlifting
Speaking a new language
You don’t need “innate talent” for any of these!
You do need (1) patience and (2) lots of practice (3) take things step by step
Speaking Stats
Speaking Stats
Treat stats (and R!) like you would learning a language
The core of generalising to new situations is grammar 😱
Generally: the rules for how to create new combinations in new situations
Not everyone’s favourite thing! But essential for fluency
Today’s lecture focuses on the “grammar” of statistics
- It’s hard work, but essential for everything that follows
Distributions
Distribution
Numerically speaking, the number of observations per each value of a variable
- Shows us which values occur more often and which less often
- The shape formed by the bars of a bar chart/histogram
A dicey example
6-sided die (unbiased)
Each value is equally likely (e.g. the probability of getting a 6 is as likely as the probability of getting a 3)
Roll it 10 times, 50 times, and 1000 times - the more roll we throw it, the more the distributions of rolls resembles the shape of the probability function
The normal distribution
Also called the Gaussian distribution, the “bell curve”
The one you need to understand
Continuous, unimodal, symmetrical, and bell-shaped
- Not every symmetrical bell-shaped distribution is normal (e.g. t)
It’s also about the proportions
- The normal distribution has fixed proportions
- A function of two parameters: \(\mu\) (mean) and \(\sigma\) (SD)
Area below the normal curve
What’s “unusual”?
- The average adult goes to 127 social events per year 👀
- The standard deviation is 40 - this is the average difference between each individual’s number of social events and the population value of 127
- This means that:
- 68% of people attend between 87 and 167 social events per year (127 \(\pm\) 40)
- 95% of people attend between 48.6 and 205.4 social events per year (127 \(\pm\) 1.96 \(\times\) 40)
What’s “unusual”?
- Meet Charlie
- Charlie goes to 57 social events per year.
- This is 70 events below the average
- Is Charlie unusual?
What’s “unusual”?
How common is Charlie’s score of 57 ?
Shaded area: proportion of the population that attends more social events than Charlie.
Non-shaded area: proportion of people who attend fewer events than Charlie
Working out proportions - the long way around
Let’s transform Charlie’s score of 57 into a Z-score:
M = 127
SD = 40
Z = (57 - 127) / 40 = -1.75
Working out proportions - the long way around
Perks of standardized scores - we know exactly how probable each Z-score is
Optional: Look up the probability of ZCharlie = -1.75 in a Z-table - e.g. https://www.z-table.com/ (or use R)
The probability of obtaining a is around 0.04 - so only 4 % of people go to fewer social events than Charlie (non-shaded area)
The area under the curve is equal to 1 - so the remaning shaded area represents 1- 0.04 = 0.96 or 96 % (people who attend more social events per year than charlie)
Working out proportions - the quickR way
Working out critical values
- Sometimes we want to find a value in a distribution that defines a cut-off point - for example top 5%.
Critical Value
A value that cuts off a specific proportion of a distribution
- How many events would Charlie need to go to, if he wanted to be among top 5% of social-event-goers on the planet?
Working out critical values
We could work backwards through Z-scores: find a Z-score that corresponds to the probability of 0.95, then transform the Z-score into the original score.
- Or use R.
Charlie would need to attend 193 social events per year (that’s 3.7 events per week! 😱 ) if he wanted to be in the top 5% of event-goers.
This is 136 more events than he attends at the moment.
One more example
Patty attends 190 events per year. Is she in the top 10% of event goers?
In a normal distribution with M = 127 (social events attended per year) and SD = 40, an individual would have to attend 178 events or more to be in the top 10% of event goers.
Therefore, with 190 events attended per year Patty is in the top 10%.
Sampling more humans
- So far, we looked at a score of one (unsociable) individual
- In research, we typically work with scores of many individuals - we collect a sample.
- Samples from the same population will be different from one another
- We can ask 6 people how many events they go to. Then ask 6 different people the next day - everyone’s score will be different.
Sampling from distributions
Statistics ( \(\bar{x}\), \(s\), etc.) of two samples will be different
Sample statistic (e.g., \(\bar{x}\)) will likely differ from the population parameter (e.g., \(\mu\))
Sampling distribution
many_sample_means <- replicate(100000, mean(rnorm(50, mean = 127, sd = 40)))
mean(many_sample_means)[1] 127.0259
The Central Limit Theorem
Take a sample
Compute the mean
Put it on the plot below
Repeat
One more peek at the dice
We know that dice rolls are uniformly distributed - each number is equally likely
What if we calculate an average roll?
One more peek at the dice
dice_rolls_6 <- replicate(10000, mean(sample(50, x = 1:6, TRUE)))
ggplot2::ggplot() +
geom_histogram(aes(x = dice_rolls_6), binwidth = 0.1, fill = "darkcyan", colour = "white") +
labs(x = "Average roll value") +
theme_minimal(base_size = 15)
The CLT governs a lot of processes where randomness and sampling are involved.
This is extremely useful for research - our tests are not immediately doomed if we collect a messy sample
NEXT WEEK
More sampling distributions
Quantifying uncertainty with standard errors and confidence intervals
